SVR dynamic system modeling with delayed output measurements

ABSTRACT

The method comprises the steps of inputting a first set of data into both a physical system and into an SVR model. The method includes collecting a second set of system data from the physical system and stacking a plurality of SVR models to form an output prediction without feeding back the model output. Another aspect of the invention includes a method of hybrid modeling having delayed output measurement. This method includes the steps of inputting a first set of data into both a physical system and into an SVR model and collecting a second set of system data from the physical system. A modeling error is injected into both the second set of system data from the physical system and a third set of system data from the SVR model wherein the modeling error thereby leads to substantially improved model output.

FIELD OF THE INVENTION

The present invention relates to nonlinear dynamic systemsidentification and modeling, in particular using SVR dynamic modelingwith delayed output measurements.

BACKGROUND OF THE INVENTION

Models of dynamic systems are of great importance in almost all fieldsof science and engineering and specifically in control, signalprocessing, and information science. Most systems encountered in thereal world are nonlinear and in many practical applications nonlinearmodels are required to achieve an adequate modeling accuracy. A model isalways only an approximation of a real phenomenon so that having anapproximation theory which allows for analysis of model quality is asubstantial concern.

A support vector regression (SVR) model was developed for accuratemodeling of nonlinear systems. While the SVR capability for representingstatic nonlinearity has been proven to be powerful in several casestudies, its application for nonlinear dynamic system modeling has metseveral challenges. Particularly, difficulty in implementing the SVRmodel in parallel configuration for dynamic systems has been identifiedas one of the key hurdles in this application. Accordingly, there existsa need in the art to provide an SVR model having a parallel orparallel-like configuration for dynamic systems.

The SVR model is optimized and implemented in a series-parallelconfiguration. FIG. 1A illustrates the series-parallel configuration.The series-parallel configuration takes the measured inputs and outputsto predict the future output of a dynamic system. The model and physicalsystem are running in parallel, but the output of the physical system isfed into the SVR model. For other applications, particularly for caseswhere outputs are not measured, a model having a parallel configuration,such as shown in FIG. 1B, is required. The parallel model does not relyon output from the physical system to predict the future output.Therefore, it can be run in parallel with the physical system.

While the derivation of the SVR models for the series-parallelimplementation can be formulated as a linear programming or quadraticprogramming optimization problem, the direct optimization of the SVRmodel for the parallel implementation is computationally prohibitive.Consequently, the derivation of the parallel model is often done as apost-processing step after the series-parallel model as shown in FIG. 1Ais derived. It is common for one to derive the series-parallel modelusing linear programming or quadratic programming and then rewrite theresulting SVR model as shown in FIG. 1A into a parallel implementationas shown in FIG. 1B.

The parallel and series-parallel models are different in structure inthat the parallel model requires internal feedback while theseries-parallel model does not. The parallel model directly translatedfrom the series-parallel model cannot be guaranteed to have the desiredproperties, such as stability and model accuracy.

SUMMARY OF THE INVENTION

The present invention provides for a method of forward prediction usingstacked SVR models. The method comprises the steps of inputting a firstset of data into both a physical system and into an SVR model. Themethod further comprises the steps of collecting a second set of delayedsystem data from the physical system and stacking a plurality of SVRmodels to form a prediction model. The resulting SVR model uses u(k−1)and delayed output y(k−d) to predict y(k) without utilizing feedbackloops from the model output thereby yielding improved accuracy comparedto the parallel SVR model. Another aspect of the invention includes amethod of hybrid modeling having delayed output measurement. This methodincludes the steps of inputting a first set of data into both a physicalsystem and into an SVR model and collecting a second set of system datafrom the physical system. A further third set of system data iscollected from the SVR model. The SVR model is provided with both thesecond set of system data from the physical system and the third set ofsystem data from the SVR model. Further, a modeling error calculated bycomparing the delayed physical output and delayed model output isinjected to form the prediction of the hybrid SVR model thereby leads tosubstantially improved model output.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates the series-parallel model of the prior art;

FIG. 1B illustrates a parallel model of the prior art;

FIG. 2 illustrates a hybrid SVR model;

FIG. 3 illustrates a building block portion of the model of the stackedSVR model;

FIG. 4 illustrates the combination of blocks made from the buildingblock as seen in FIG. 3;

FIG. 5 illustrates a final composite SVR model using u(k−1) and y(k−3)without feedback loops;

FIG. 6 illustrates a hybrid SVR model with error injection;

FIG. 7 illustrates a graphical comparison of different “hybrid” modelsfor rbf kernel SVR model;

FIG. 8 illustrates a graphical representation and comparison ofdifferent “hybrid” models for the tensor wavelet kernel SVR model; and

FIG. 9 illustrates a graphical comparison of different “hybrid” modelsfor the raised cosine wavelet kernel SVR model.

DETAILED DESCRIPTION OF THE INVENTION

The present invention and model incorporates delayed SVR series-parallelstructure as the submodels and uses a forward path to construct theinformation needed for output prediction thereby avoiding feedback loopsin the parallel model thereby avoiding ill behavior and instability. Analternative approach is to inject the delayed output and thecorresponding measurable model error into the parallel model andmitigate the drawbacks of the parallel SVR model without incurringsubstantial additional computation.

A support vector machine allows modeling of complex dynamic systems. Thekernel of the support vector machine may be various types. In examplesof the present invention, as pertaining to the present example asapplied to a robot arm system and data, the example assumes y(k−3) isavailable, but the structure is general, and can be applied to caseswhen y(k−2) or y(k−5) is available rather than y(k−3). The present SVRmodel is applied to a robot arm system. The modeling using supportvector machine is illustrated with a wavelet kernel, which is wellsuited to modeling nonlinear systems. These kernels are used inconnection with several SVR models. Improvement in model accuracy isobtained by (a) using composite kernels (the ARMA2k model) and (b)properly constructing new kernel functions. In particular, the use of“raised cosine wavelet” kernel provides the most desirable results. Inthe present example, we will consider the following baseline ARMA2kmodel

$\begin{matrix}{{y(k)} = {\sum\limits_{\beta_{i}}\;\left. 〚{{\left( k〛 \right._{1\; i}\left( {v_{1},{sv}_{1\; i}} \right)} + {k_{2\; i}\left( {v_{2},{sv}_{2\; i}} \right)}} \right)}} & (1)\end{matrix}$where v₁=[u(k−1), u(k−2)], v₂=[y(k−1), y(k−2), y(k−3)], are thecorresponding moving average (MA) and auto-regression (AR) regressorsfor the ARMA2k model. sv_(1i), sv_(2i), i=1, 2 . . . are the supportvectors obtained from linear programming optimization process. The modelcan be put in the general format asŷ _(ps)(k)=f(u(k−1),u(k−2),y(k−1),y(k−2),y(k−3))  (2)if it is implemented in the series-parallel form orŷ _(p)(k)=f(u(k−1),u(k−2),ŷ _(p)(k−1), y _(p)(k−2),ŷ _(p)(k−3))(2)  (3)if it is implemented in the parallel form.

Applications include all types of nonlinear system modeling, and thegeneral approach described has widespread applications includingelectronic engine control in automobiles, image compression, speechsignal processing, process control generally, and other complexexamples.

To evaluate the effects of using delayed output in the “hybrid” SVRmodel three different SVR baseline models are considered. First, rbfkernels. Both the MA and the AR kernels are rbf function of the form:

$\begin{matrix}{{k_{1,2}\left( {x,y} \right)} = {{\mathbb{e}}^{\frac{{- {({x - y})}} \cdot {({x - y})}^{1}}{2\; p^{2}}}\mspace{14mu}{for}\mspace{14mu}{some}\mspace{14mu}{parameter}\mspace{14mu}{p.}}} & (4)\end{matrix}$Second, the tensor wavelet kernels wherein the hybrid spline function isused for the MA kernel and the tensor wavelet is used for the AR kernel

$\begin{matrix}{\mspace{79mu}{{{k_{1}\left( {x,y} \right)} = {\exp\left( {local}_{kernel} \right)}},\mspace{20mu}{{{local}_{kernal}\left( {x,y} \right)} = {\exp\left( {- \sqrt{\left( {\left( {x - y} \right)*\frac{\left( {x - y} \right)^{1}}{2\; p^{2}}} \right)}} \right)}}}} & (5) \\{{{k_{2}\left( {x,y} \right)} = {z_{1}z_{1}\mspace{14mu}\ldots\mspace{20mu} z_{n}}},{z_{i} = {{\cos\left( {p_{1\; i}\left( {{x(i)} - {y(i)}} \right)} \right)}*{\exp\left( {- \frac{\left( {{x(i)} - {y(i)}} \right)^{2}}{2\; p_{zi}}} \right)}}}} & (6)\end{matrix}$and third, the raised cosine wavelet kernel wherein the third-orderB-spline function is used for the MA kernel and the raised cosinewavelet kernel is used for the AR kernel. The B-spline function has theform

$\begin{matrix}{\mspace{79mu}{{{k_{1}\left( {x,y} \right)} = {z_{11}z_{12}\mspace{14mu}\ldots\mspace{14mu} z_{1\; n}}},\mspace{31mu}{z_{1\; i} = {B_{3}\left( {x_{i} - y_{i}} \right)}}}} & (7) \\{{B_{3}( \cdot )} = {\frac{1}{3!}\left( {{\begin{pmatrix}4 \\0\end{pmatrix}\left( {\cdot {+ 2}} \right)_{+}^{3}} - {\begin{pmatrix}4 \\1\end{pmatrix}\left( {\cdot {+ 1}} \right)_{+}^{3}} + {\begin{pmatrix}4 \\2\end{pmatrix}( \cdot )_{+}^{3}} - {\begin{pmatrix}4 \\3\end{pmatrix}\left( {\cdot {- 1}} \right)_{+}^{3}} + {\begin{pmatrix}4 \\4\end{pmatrix}\left( {\cdot {- 2}} \right)_{+}^{3}}} \right)}} & (8)\end{matrix}$and (z)₊=max(z,0). The raised cosine wavelet kernel takes the form of

$\begin{matrix}{\mspace{79mu}{{{k_{2}\left( {x,y} \right)} = {z_{21}z_{22\mspace{14mu}}\ldots\mspace{14mu} z_{2n}}},\mspace{31mu}{= {{\frac{n_{i\; 1}}{d_{i\; 1}} - {\frac{n_{i\; 2}}{d_{i\; 2}}n_{i\; 1}}} = {{{\sin\left( \frac{2{\pi\left( {1 - p_{1}} \right)}\left( {{x(i)} - {y(i)} - p_{2}} \right)}{p_{3}} \right)} + {{\sin\left( \frac{2{\pi\left( {1 + p_{1}} \right)}\left( {{x(i)} - {y(i)} - p_{2}} \right)}{p_{3}} \right)}n_{i\; 2}}} = {{{\sin\left( \frac{2{\pi\left( {1 - p_{1}} \right)}\left( {{x(i)} - {y(i)} - p_{2}} \right)}{p_{3}} \right)} + {{\sin\left( \frac{2{\pi\left( {1 + p_{1}} \right)}\left( {{x(i)} - {y(i)} - p_{2}} \right)}{p_{3}} \right)}\mspace{20mu} d_{i\; 1}}} = {{2{\pi\left( \frac{{x(i)} - {y(i)} - p_{2}}{p_{3}} \right)}\left( {1 + {4\; p_{1}\frac{{x(i)} - {y(i)} - p_{2}}{p_{3}}}} \right)\mspace{20mu} d_{i\; 2}} = {2{\pi\left( \frac{{x(i)} - {y(i)} - p_{2}}{p_{3}} \right)}\left( {1 - {2\; p_{1}\frac{{x(i)} - {y(i)} - p_{2}}{p_{3}}}} \right)}}}}}}}} & (9)\end{matrix}$where p's are the kernel parameters that have different interpretationfor different kernels.

The results of three various models for the above referencedseries-parallel and parallel implementations applied to the robot armbenchmark problem are summarized in Table 1 as the baseline foranalysis. The rbf has the simplest model expression, but its parallelimplementation has the largest error. The raised cosine wavelet kernelhas the best parallel implementation results.

TABLE 1 rms error of the SVR models tested on the robot arm benchmarkdata set SVR Model kernel Parallel-series Parallel Rbf 0.1498 0.9348Tensor Wavelet 0.0965 0.5954 Raised Cosine 0.1423 0.5419

Example 1 Hybrid Model with a Delayed Output Insertion

We will assume y(k) is measured, but is available only after some delay.Therefore, it is assumed that y(k−3) is available for the robot armproblem. Utilizing the series-parallel and parallel models shown inFIGS. 1A and 1B, one straightforward approach is to incorporate themeasured output y(k−3) wherever ŷ_(p)(k−3) is needed in the parallelmodel. This embodiment is shown in FIG. 2. The model in FIG. 2 is the“direct hybrid” SVR model. The direct hybrid SVR model is expressed asŷ _(h)(k)=f(u(k−1),u(k−2),ŷ _(h)(k−1),ŷ _(h)(k−2),y(k−3))  (10)The model above is straightforward and intuitive but is not a viablemodel and does not improve the parallel model performance. Analyticallyshown in Table 2 below a physical system can become unstable when it isturned into the model of the direct hybrid as shown in formula (10)above. The numerical case studies are summarized below in Table 2. Datafrom the various outputs and inputs are compared in order to achieve themost accurate models. Output data from the physical system is comparedto output data from the model system.

TABLE 2 rms error of the “direct hybrid” SVR models tested on the robotarm benchmark data set SVR Model kernel Parallel-series Parallel DirectHybrid Rbf 0.1498 0.9348 2.4503 Tensor Wavelet 0.0965 0.5954 UnboundedRaised Cosine 0.1423 0.5419 1.8316As such, direct insertion of the delayed output does not provideimprovement in any of the three SVR baseline models. Accordingly, thereexists a need in the art to provide an improved model.

Example 2 Forward Prediction Using Stacked SVR Models

The present embodiment is based on stacking multiple series-parallelmodels. This model incorporates delayed SVR series-parallel structure asthe submodels and uses a forward path to construct the informationneeded for output prediction, thereby avoiding the feedback loop in theparallel model providing for improved stability and prediction accuracy.

The stacked series-parallel model includes the series-parallel model asa building block. The series-parallel model is illustrated in FIG. 1A.The prediction of y(k) requires y(k−1), y(k−2) and y(k−3), but onlyy(k−3) is available. Rather than feeding back the model output fory(k−1) and y(k−2), the delayed input and output and the series-parallelmodel is used to construct the other signals to replace y(k−1) andy(k−2). The proposed stacked model is expressed in the form ofŷ _(s)(k)=f(u(k−1),u(k−2), y (k−1), y (k−2),y(k−3))(4a)y (k−1)=f(u(k−2),u(k−3), y (k−2),y(k−3),y(k−4))(4b)y (k−2)=f(u(k−3),u(k−4),y(k−3),y(k−4),y(k−5))(4c)  (11)The same function f is applied three times in calculating ŷ_(s)(k), butthe value of ŷ_(s)(k) is not fed back or reused in generating futureprediction. Therefore, all calculations in (11) are in the forwarddirection and the model (11) does not contain any feedback loop such asin a parallel model. As such, the properties of the stacked SVR modelare much like those of the series-parallel model. The modelimplementation is represented in the flowcharts as seen in FIGS. 3-5.Data from the various outputs and inputs are compared in order toachieve the most accurate models. Output data from the physical systemis compared to output data from the model system.

FIG. 3 illustrates the building block of the basic series-parallelmodel. Input to the physical system u(k) results in an output of y(k).The input u(k) and output y(k) of the physical system as well as theirdelayed versions are inserted into the SVR model. FIGS. 4 and 5illustrate the stacking of the building blocks as shown in FIG. 3. FIG.4 illustrates the blocks made from the building block as seen in FIG. 3to generate y(k−2) and y(k−1). FIG. 5 illustrates the final compositeSVR model using u(k−1) and y(k−3) without any feedback loops. Theperformance of the stacked model is evaluated for three different SVRmodels. The results of this evaluation are shown below in Table 3. Allthree hybrid models yield excellent accuracy and are a substantialimprovement compared to their parallel counterparts. The number of submodels used for the stacked SVR model varies depending on an outputdelay.

TABLE 3 rms error of the “stacked” SVR models tested on the robot armbenchmark data set Stacked Feedforward SVR Model kernel Parallel-seriesParallel Model Rbf 0.1498 0.9348 0.3266 Tensor Wavelet 0.0965 0.59540.2267 Raised Cosine 0.1423 0.5419 0.2931

Example 3 Hybrid SVR Model with Error Injection

An alternative embodiment and approach includes the injection of thedelayed output y(k−3) and the corresponding measurable modeling errorinto the parallel model to mitigate the drawbacks of the parallel SVRmodel while avoids additional computation of the stacked SVR modeldiscussed above.

Instead of the direct hybrid model as shown and discussed above and informula (11), a hybrid model is utilized with an error injection in theform ofŷ _(hp)(k)=f(u(k−1),u(k−2),ŷ _(hp)(k−1),ŷ _(hp)(k−2),y((k−3))−c*(ŷ_(hp)(k−3)−y(k−3))  (12)where c*(ŷ_(hp)(k−3)−y(k−3)) is the extra term that injects the modelingerror to the equation to help stabilize the model, and c is the designparameter that can be tuned for desired performance. A graphical andflowchart representation of the present model is illustrated in FIG. 6.Data from the various outputs and inputs are compared in order toachieve the most accurate models. Output data from the physical systemis compared to output data from the model system.

FIG. 6 illustrates the parallel SVR model having delayed outputmeasurement and error injection. The result of applying the presenthybrid model to the robot arm benchmark data is summarized below inTable 4. When compared to the parallel-series, parallel, and directhybrid model, the results indicate the performance as shown in Table 4.The value of the c parameter, which is adjusted to achieve the desiredmodeling accuracy, is shown in the table for various SVR models.

TABLE 4 rms error of the hybrid SVR models with error injection, testedon the robot arm benchmark data set Hybrid SVR Model Parallel- Directw/error c- kernel series Parallel hybrid injection parameter Rbf 0.14980.9348 2.4503 0.2213 1 Tensor 0.0965 0.5954 Unbounded 0.1980 0.7 WaveletRaised 0.1423 0.5419 1.8316 0.2107 1 CosineFIGS. 7-9 illustrate detailed responses of various model configurationsas discussed above. The original SVR models are derived using the first500 points as training data. The traces given in the plots as shown inFIGS. 7-9 illustrate both the training data and the validation data (thelast 520 points) in the results.

Applications of examples of the present invention include modeling ofdynamic systems, such as process control, including chemical, physical,and/or mechanical processes control. Further applications include imageanalysis (such as image cataloging, object recognition such as facedetection, and the like), voice recognition, and the computationallydemanding other applications. A particular application is enginemanagement and control applications, described further below. Vehiclessuch as automobiles often include an electronic control unit (ECU) forengine control. The ECU receives engine data from various enginesensors, using the engine data and an engine model to modify the engineoperational parameters. Engine sensors providing engine data may includea throttle position sensor, oxygen sensor, manifold absolute pressuresensor, air temperature sensor, coolant temperature sensor, and thelike. Engine parameters controlled by an ECU may include fuel injectionquantities, ignition timing, variable valve timing, turbocharger boost,and the like. In a typical example, the throttle position sensorprovides a throttle position signal as part of the engine data receivedby the ECU, which in response increases the fuel injection quantities.

Engine operation is enhanced by using by using an ECU that uses anaccurate model of engine operation. However, the number of modelparameters may become large, so that the computing power necessary toset up and run the model becomes substantial, particularly usingconventional approaches. Engine operation is a dynamic process andresponse to transient data may need to be substantially immediate.Operator inputs are received by the ECU, and result in control of engineoperation parameters (such as fuel injection quantity). There also maybe adverse safety consequences if an engine model used does not readilyallow the ECU to detect or respond quickly to transient data. Hence,dynamic modeling methods using models described herein are particularlyuseful for engine control applications.

For a new vehicle engine, model parameters may be determined from avarious engine testing and model training process using another of thesame type of engine. However, these training parameter values may becomemore inaccurate as the engine wears. Hence, the dynamic model can beadjusted based on a comparison of model predictions and actual observedengine performance. This allows individual optimization of engineperformance. The optimized model parameters may further be diagnostic ofengine condition.

A support vector machine can be used to model any dynamic process, andis particularly well suited to engine management applications. Inexamples of the present invention, dynamic system modeling using asupport vector machine uses a computationally simple linear programmingapproach.

Embodiments of the present invention allow improved engine behaviormodeling as a function of adjustable engine parameters. An ECU for anengine can use the models discussed above to improve engine performance.Examples include improved engine modeling to allow optimization inexhaust emission reduction, fuel economy enhancement, acceleration, orother engine performance metrics.

Conventional approaches become computationally challenging as the numberof engine control inputs increase. Further, conventional modelingapproaches such as the use of lookup tables are not adaptive, often usesteady-state operation to generate the model, and often cannot properlydescribe transient behavior of the engine. Conversely, engine modelingaccording to the present embodiment provides accurate representation ofengine performance data. Further, training of an engine model accordingto the present invention may be much simpler.

Embodiments of the present invention may be used for other types ofmodeling, in particular modeling of any type of complex multi input,multi output nonlinear systems. Sparsity of the generated modelfacilitates real-time computation and online model adaptability.Embodiments of the present invention include modeling of transientresponses in complex systems. The present invention is not restricted tothe illustrative examples and embodiments described above. The examplesare not intended as limitations on the scope of the invention. Methods,apparatus, and the like described herein are exemplary and not intendedas limitations on the scope of the invention. Changes therein and otheruses will occur to those skilled in the art. The scope of the inventionis defined by the scope of the appended claims.

Having described our invention, we claim:
 1. A method of forwardprediction using stacked SVR models for control of an engine, the enginehaving an Electronic Control Unit (ECU), the method comprising the stepsof: inputting a first set of data u(k) into both a physical system andinto an SVR model, the first set of data u(k) received from at least oneengine sensor; collecting a second set of system data y(k) from thephysical system; stacking a plurality of SVR models, the stacked SVRmodel having the formŷ _(s)(k)=f(u(k−1),u(k−2), y (k−1), y (k−2),y(k−3))y (k−1)=f(u(k−2),u(k−3), y (k−2),y(k−3),y(k−4))y (k−2)=f(u(k−3),u(k−4),y(k−3),y(k−4),y(k−5)) the resulting stacked SVRmodel using u(k−1) and y(k−3) without using feedback loops yieldingimproved accuracy, the ECU configured to modify operation parameters ofthe engine.
 2. The method in accordance with claim 1 further includingthe step of applying the same function f three times in calculatingŷ_(s)(k)=f(u(k−1),u(k−2), y(k−1), y(k−2),y(k−3)) wherein the value ofŷ_(s)(k)=f(u(k−1),u(k−2), y(k−1), y(k−2),y(k−3)) is not fed back intothe stacked SVR model.
 3. The method of claim 1 further including thestep of comparing data of the physical system to data resulting from themodel.
 4. A method of hybrid modeling with delayed output measurementsfor control of an engine, the engine having an Electronic Control Unit(ECU), the method comprising the steps of: inputting a first set of datau(k) into both a physical system and into an SVR model, the first set ofdata u(k) received from at least one engine sensor; collecting a secondset of system data y(k) from the physical system; collecting a third setof system data from the SVR model; providing the SVR model with both thesecond set of system data from the physical system and the third set ofsystem data from the SVR model; injecting a modeling error into both thesecond set of system data from the physical system and the third set ofsystem data from the SVR model; modeling the SVR model, the SVR modelhaving the formŷ _(hp)(k)=f(u(k−1),u(k−2),ŷ _(hp)(k−1),ŷ _(hp)(k−2),y((k−3))−c*(ŷ_(hp)(k−3)−y(k−3)) wherein −c*(ŷ_(hp)(k−3)−y(k−3)) is the modeling errorthereby leading to an improved model output wherein the physical systemis changed based on the SVR model, the ECU configured to modifyoperation parameters of the engine.
 5. The method of claim 4 furtherincluding the step of comparing data of the physical system to dataresulting from the model.
 6. A method of forward prediction of aphysical system using stacked SVR models, the physical system having anElectronic Control Unit (ECU), the method comprising the steps of:inputting a first set of data u(k) into both a physical system and intoan SVR model, the first set of data u(k) received from at least onesensor; collecting a second set of system data y(k) from the physicalsystem; stacking a plurality of SVR models wherein an output data is fedinto the next stacked SVR model; collecting the output data from boththe physical system and the stacked SVR models, comparing the outputdata from both the physical system and the stacked SVR models theresulting stacked SVR model using u(k−1) and delayed y(k-d) withoutusing feedback loops yielding improved accuracy, the ECU configured tomodify operation parameters of the physical system.
 7. The method ofclaim 6 wherein the number of SVR models used for the stacked SVR modelvaries depending on an output delay.
 8. The method in accordance withclaim 6 wherein the stacked SVR model has the form:ŷ _(s)(k)=f(u(k−1),u(k−2), y (k−1), y (k−2),y(k−3))y (k−1)=f(u(k−2),u(k−3), y (k−2),y(k−3),y(k−4))y (k−2)=f(u(k−3),u(k−4),y(k−3),y(k−4),y(k−5)).
 9. The method inaccordance with claim 8 further including the step of applying the samefunction f three times in calculating ŷ_(s)(k)=f(u(k−1),u(k−2), y(k−1),y(k−2),y(k−3)) wherein the value of ŷ_(s)(k)=f(u(k−1),u(k−2), y(k−1),y(k−2),y(k−3)) is not fed back into the stacked SVR model.